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Theorem sspwimpALT2 41559
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpALT2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3483 . . . 4 𝑥 ∈ V
2 elpwi 4531 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 id 22 . . . . 5 (𝐴𝐵𝐴𝐵)
42, 3sylan9ssr 3967 . . . 4 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
5 elpwg 4525 . . . . 5 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
65biimpar 481 . . . 4 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
71, 4, 6sylancr 590 . . 3 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
87ex 416 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
98ssrdv 3959 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  Vcvv 3480  wss 3919  𝒫 cpw 4522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524
This theorem is referenced by: (None)
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